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Full text of "Treatise On Analysis Vol-Ii"

21 PRODUCT OF MEASURES 227 with the product convention referred to above. Hence it is enough to prove the inequality (13.21.11.2) JJ Now this inequality is clearly valid when the right-hand side is equal to -f oo. So consider the case in which each of the factors on the right is finite. Then there exist two decreasing sequences (/„), (#„) such that fn e ./(X), )9f^fH9ff^gH for all n, and * „ dp. r* r* r* /* /Ai=lim /ndA, #^ = lim J n~xx> J J />-+oo J By reason of our conventions and of (12.7.5), the function (x, y)*~~*fn(x)gn(y) belongs to */(X x Y), and we have f(x)g(y) ^fn(x)gn(y) for all n and all (x9 j) e X x Y. But, by virtue of (13.21.3), f * J - ff* JJ "X9 from which the desired inequality follows by letting n tend to +00. Finally, to deal with the case where (for example) / is ^-negligible, it is enough (by virtue of our conventions) to prove that the function (x,y)\-+f(x)g(y) is v-negligible. This is a consequence of the following proposition: (13.21.12) If N is any ^-negligible subset of X, then the set N x Y is v- negligible. For since Y is the union of a denumerable sequence (Ln) of compact sets, it is enough to show that each of the sets N x Ln is v-negligible. But since we have proved that (13.21.11.1) is valid whenever both of the factors on the right-hand side are finite, we may take/= <pN and g = <pLn in this formula. (13.21.13) Let E, F, G be three topological spaces and u : E x F -» G a con- tinuous mapping. L#//(resp. g) be a ^-measurable mapping ofX into E (resp. a ^-measurable mapping ofY into F). Then (x, y)^u(f(x),g(yj) is a v-measur- able mapping ofX x Y into G. ' J